1.
\(\displaystyle \int x^3\ln(x)\, dx\)
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Worksheet Some Exercises for this section
I included some practice problems that cover some main concepts in this section. You don’t need to turn it in, but I highly encourage you to work on this with your classmates. I may take problems here to be your in-class practice problems, homework problems, and/or exam problems. Reach out to Richard for help if you get stuck or have any questions.
I will only include the final answer to some of the problems for you to check your result. If you want to check your work, talk to Richard and he is happy to discuss the process with you.
Exercise Group.
Evaluate the following indefinite integral using integration by parts.
2.
\(\displaystyle \int \frac{\ln(x)}{x^5}\, dx\)
3.
\(\displaystyle \int \arccos(x)\, dx\)
4.
\(\displaystyle \int e^{3x}\cos(4x)\, dx\)
Exercise Group.
Evaluate the following indefinite integral using multiple methods indicated in the hint below.
P.S.: I encourage you to try evaluating the following integral in a different way (be rebellious and adventurous a bit). We will talk about strategies in evaluating integrals in Section 7.6 where one of the important idea is that there may be multiple strategies to evaluate an integral.
5.
\(\displaystyle \int \left(\arcsin(x)\right)^2\, dx\)
6.
\(\displaystyle \int \frac{\left(\ln(x)\right)^2}{x^2}\, dx\)
Exercise Group.
Evaluate the following definite integral. Make sure your notation is clear and correct.
7.
\(\displaystyle \int_0^{\frac{\pi}{4}} x\sin(2x)\, dx\)
8.
\(\displaystyle \int_0^3 \ln \left(x^2 + 1\right)\, dx\)
